Question

A gas sample occupying a volume of \(25.6 \mathrm{~mL}\) at a pressure of 0.970 atm is allowed to expand at constant temperature until its pressure reaches 0.541 atm. What is its final volume?

Short answer

The final volume is approximately 45.9 mL.

Step-by-step solution

Identify the Given and Required Information

We are given the initial volume \( V_1 = 25.6 \; \mathrm{mL} \), the initial pressure \( P_1 = 0.970 \; \mathrm{atm} \), and the final pressure \( P_2 = 0.541 \; \mathrm{atm} \). We need to find the final volume \( V_2 \).

Apply Boyle's Law

According to Boyle's Law, for a constant temperature, the product of pressure and volume is constant: \( P_1 \times V_1 = P_2 \times V_2 \). We can use this relationship to find the unknown final volume \( V_2 \).

Rearrange Boyle's Law to Solve for Final Volume

Solve for \( V_2 \) by rearranging the equation: \[ V_2 = \frac{P_1 \times V_1}{P_2} \].

Substitute the Known Values into the Equation

Insert the known values into the rearranged equation:\[ V_2 = \frac{0.970 \times 25.6}{0.541} \].

Calculate the Final Volume

Perform the calculation:\[ V_2 = \frac{24.832}{0.541} \approx 45.9 \; \mathrm{mL} \].

Key concepts

Gas Laws

Gas laws are fundamental principles in chemistry that relate the properties of gases, such as pressure, volume, and temperature. These laws allow us to predict and understand how gases behave under various conditions. Boyle's Law, Charles's Law, and Avogadro's Law are some of the main gas laws which help describe these relationships. For instance, Boyle's Law specifically deals with how pressure and volume of a gas interact while keeping the temperature constant. When studying gas laws, we assume gases exhibit ideal behavior, meaning they follow these laws perfectly under normal conditions.

• Boyle's Law: Describes the pressure-volume relationship at constant temperature.
• Charles's Law: Relates volume and temperature, holding pressure constant.
• Avogadro's Law: Connects volume and quantity of gas molecules, assuming constant temperature and pressure.

Understanding these laws is crucial for predicting how changes in one property of a gas will affect the others. It's important to remember that while these laws are accurate under many conditions, real gases might diverge slightly due to non-ideal interactions.

Pressure-Volume Relationship

The pressure-volume relationship of gases, as outlined in Boyle’s Law, demonstrates that the pressure of a gas is inversely proportional to its volume, provided the temperature remains constant. This principle means that if you decrease the volume of a gas, its pressure increases, and vice versa.

• Mathematically, this relationship is represented as: \( P_1 \times V_1 = P_2 \times V_2 \).
• This means when one multiplies the initial pressure and volume, it should be equal to the product of the final pressure and volume.

In practical terms, imagine a balloon: squeezing it reduces the volume inside, which, according to Boyle's Law, increases the pressure the gas exerts internally.

It's also important to note that this relationship assumes no other factors alter the gas's behavior, such as temperature or external pressures. This direct relationship assists in many real-world applications, from understanding balloons to optimizing engines.

Ideal Gas Behavior

Ideal gas behavior is an assumption where we consider gas molecules to have perfectly elastic collisions and no volume or intermolecular forces. This simplification allows for the easy application of gas laws. In reality, ideal gases don't exist, but many gases behave almost ideally under typical conditions like standard temperature and pressure (STP).

• Ideal gases obey all gas laws precisely, making calculations straightforward.
• Real gases show slight deviations, especially at extremely high pressures or low temperatures due to intermolecular attractions and volumes.

In the exercise solution, we assumed ideal gas behavior to apply Boyle’s Law effectively. This is a valid approximation in most scenarios, as the small deviations typically do not significantly affect outcomes at common conditions. Maintaining this assumption simplifies solving gas-related problems, ensuring the results remain reasonably accurate.